Vz-GAL Dusty Star-Forming Galaxies: Revisiting the CO-H2 Conversion Factor Tension

The CO luminosity-to-H$_2$ mass conversion factor ($\alpha_{CO}$) remains a debated uncertainty in determining molecular gas masses of high-redshift dusty star-forming galaxies (DSFGs). Dynamical mass constraints have often favored $\alpha_{CO}=0.8$~$M_{\odot}~{(K~km~{s}^{-1}~{pc}^{2})}^{-1}$, whereas dust- and radiative-transfer-based methods imply higher values. We revisit this ``tension" using the largest homogeneous sample of 21 unlensed $z\sim1-4$ DSFGs, with securely measured \coonezero luminosities from the VLA \vzgal survey and resolved ($\sim{0.1}^{\prime\prime}$) ALMA 1~mm dust continuum imaging. For 12 galaxies with robust modeling constraints, we derive molecular gas masses using dust spectral energy distribution modeling and the TUNER LVG framework, adopting a solar-metallicity gas-to-dust mass ratio of 100. Although not fully independent due to shared assumptions on dust properties, these approaches yield mutually consistent gas masses corresponding to $\alpha_{CO}\sim1.5-11.5$, with a median near the Galactic $\alpha_{CO}=4.3$. Isotropic virial dynamical masses agree with these gas masses when realistic molecular gas sizes are adopted, while our proposed ``mixed" (rotating, pressure-supported, thick-disk) estimator systematically underestimates dynamical masses, producing low $\alpha_{CO}$ limits. Using GN20 ($z=4.055$) as a case study, we show that resolved gas geometry and kinematics reconcile the discrepancy with LVG-derived $\alpha_{CO}$. Our results suggest that current data do not require $\alpha_{CO}=0.8$, and intermediate to near-Galactic values remain dynamically viable given uncertainties in gas geometry, dust properties, and gas-to-dust ratios. Further progress in calibrating $\alpha_{CO}$ in the early universe will require resolved molecular gas observations, physically motivated ISM modeling, and stringent constraints on dust properties.